Non-Newtonian fluids are notoriously difficult to treat analytically and computationally. When in addition the flow is constrained by the very complex boundary conditions of the porous medium, the effective rheology of the fluid flow is not well understood.
This might for example be seen in the fact that the leading theory for describing immiscible multi-phase flow in porous media is still the relative permeability theory dating from 1936 [1]} a theory which has evident faults.
